SECTION I.

FUNDAMENTAL RULES OF ARITHMETIC.

THESE are four, ADDITION, SUBTRACTION, MULTIPLICATION, and DIVISION; they may be either simple or compound; simple, when the numbers are all of one sort or denomination; compound, when the numbers are of different denominations.

THEY are called, Principal or Fundamental Rules, because all other rules and operations in arithmetic are nothing more than various uses and repetitions of these four rules.

The object of every arithmetical operation, is, by certain given quantities which are known, to find out others which are unknown. This cannot be done but by changes effected on the given numbers; and as the only way in which numbers can be changed is either by increasing or diminishing their quantities, and as there can be no increase or diminution of numbers but by one or the other of the above operations, it consequently follows, that these four rules embrace the WHOLE art of Arithmetic.

1. SIMPLE ADDITION.

SIMPLE ADDITION is the putting together of two or more numbers, of the same denomination, so as to make them one whole or total number, called the sum, or amount.

RULE.

1. Place the numbers to be added one under another, with units under units, tens under tens, &c. and draw a line under the lowest number.

2. Add the right hand column, and if the sum be under ten, write it under the column; but if it be ten, or any exact number of tens, write a cypher; and if it be not an exact number of tens, write the excess above tens at the foot of the column, and for every ten the sum contains, carry one to the next column, and add it in the same manner as the former.

3. Proceed in like manner to add the other columns, carrying for the tens of each to the next, and set down the full sum of the last or left hand column.

PROOF.

Reckon the figures from the top downwards, and if the work be right, this amount will be equal to the first ;-or, what is often practised," cut "off the upper line of figures and find the amount of the rest; then if the "amount and upper line when added, be equal to the sum total, the work "is supposed to be right."

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To find the answer or amount of the sums given to be added, begin with the right hand column, and say, 3 and 1 make 4, and 3 are 7, and 2 are 9; which sum (9) being less than ten, set down directly under the column added. Then proceeding to the next column, say again; 5 and 4 are 9, and 1 is 10; being even ten, set down 0, and carry one to the next column, saying 1, which I carry to 6 is 7, and 0 is nothing, but 6 make 13; which sum (13) is an excess of 3.over even ten; therefore set down 3 and carry 1 for the ten to 3 in the next column, saying 1 to 8 is 9, and 3 are 12; this being the last column, set down the whole number (12) placing the 2, or unit figure directly under the column, and carrying the other figure, or the 1, forward to the next place on the left hand, or to that of Tens of thousands, and the work is done.

Ir may now be required to know if the work be right. To exhibit the method of proof let the upper line of figures be cut off as seen in the example. Then adding the three lower lines which remain, place the amount (8637) under the amount first obtained by the addition of all the sums, observing carefully that each figure fall directly under the column which produced it; then add this last amount to the upper line which you cut off; thus 7 to 2 are 9; 9 to 1 are 10; carry one to 6 is 7 and 6 are 13; 1 which I carry again to 8 is 9 and 3 are 12, all which being set down in their proper places, and as seen in the example, compare the amount. (12309) last obtained, with the first amount (12309) and if they agree, as it is seen in this case they do, then the work is judged to be right.

Note.-THE reason of carrying for ten in all simple numbers is evident from what has been taught in Notation. It is because 10 in an inferior column is just equal in value to 1 in a superior column. As if a man should

be holding in his right hand half pistareens, and in his left, dollars. If you should take 10 half pistareens from his right hand, and put one dollar into his left hand, you would not rob the man of any of his money, because 1 of those pieces in his left hand is just equal in value to 10 of those in his right hand,

SUPPLEMENT TO ADDITION.

THE attentive scholar who has understood, and still carries in his mind, what has already been taught him of Addition, will be able to answer his instructor to the following

1. What is simple Addition?

QUESTIONS.

2. How do you place numbers to be added?

3. Where do you begin the addition?

4. What is the answer called?

5. How is the sum or amount of each column to be set down?

16. What do you observe in regard to setting down the sum of the last column?

7. Why do you carry for ten rather than any other number?

2. How is addition proved?

NOTE. Should the learner find any difficulty in giving an answer to the above questions, he is advised to turn back and consult his Rule, with its illustrations.

EXERCISES.

1. What is the amount of 2801 dollars; 765 dollars; and of 397 dollars, when added together? Ans. 3963 dollars.

2. Suppose you lend a neighbour £210 at one time, £76 at another, €17 at another, and £9 at another. What is the sum lent? Ans. £312.

NOTE. The scholar who looks at greatness in his class, will not be dis'couraged by a little difficulty which may at first occur in stating his question, but will apply himself the more closely to his Rule, and to thinking, that if possible he may be able himself to answer what another may, be obliged to have taught him by his instructor.

3. A tree was broken off by the wind, 27 feet from the ground; the part broken off was 71 feet long; what was the height of that tree before it was broken?

Ans. 93 feet.

4. A man being asked his age, said he was 27 years old when he married, and he had been married 15 years. What was the man's

age?

Ans. 42.